Symmetry 2011, 3, 365-388; do:10.3390/sym3020365 OPEN ACCESS symmetry ISSN 2073-8994 www.mdp.com/journal/symmetry Artcle Any Par of 2D Curves Is Consstent wth a 3D Symmetrc Interpretaton Tadamasa Sawada *, Yunfeng L and Zygmunt Pzlo Department of Psychologcal Scences, Purdue Unversty, West Lafayette, IN 47907-2081, USA; E-Mals: l135@purdue.edu (Y.L.); pzlo@psych.purdue.edu (Z.P.) * Author to whom correspondence should be addressed; E-Mal: sawada@psych.purdue.edu; Tel.: +1-765-494-6865; Fax: +1-765-496-1264. Receved: 10 February 2011; n revsed form: 27 May 2011 / Accepted: 30 May 2011 / Publshed: 10 June 2011 Abstract: Symmetry has been shown to be a very effectve a pror constrant n solvng a 3D shape recovery problem. Symmetry s useful n 3D recovery because t s a form of redundancy. There are, however, some fundamental lmts to the effectveness of symmetry. Specfcally, gven two arbtrary curves n a sngle 2D mage, one can always fnd a 3D mrror-symmetrc nterpretaton of these curves under qute general assumptons. The symmetrc nterpretaton s unque under a perspectve projecton and there s a one parameter famly of symmetrc nterpretatons under an orthographc projecton. We formally state and prove ths observaton for the case of one-to-one and many-to-many pont correspondences. We conclude by dscussng the role of degenerate vews, hgher-order features n determnng the pont correspondences, as well as the role of the planarty constrant. When the correspondence of features s known and/or curves can be assumed to be planar, 3D symmetry becomes non-accdental n the sense that a 2D mage of a 3D asymmetrc shape obtaned from a random vewng drecton wll not allow for 3D symmetrc nterpretatons. Keywords: 3D symmetry; 3D recovery; 3D shape; degenerate vews; human percepton
Symmetry 2011, 3 366 1. Introducton Curves n a 2D mage provde very effectve nformaton about the 3D shape out there [1]. Fgure 1 shows a smple example. The reader can easly see the 3D shape of the closed contour even though the mage tself s 2D. Demo 1 [2] llustrates the 3D nterpretaton that agrees wth what the reader perceves by lookng at Fgure 1. The 3D recovery shown n the Demo used 3D symmetry as a constrant. Ths nformal observaton s consstent wth results of a number of psychophyscal experments. Specfcally, t has been shown that humans can perceve 3D shapes of objects and recognze the objects as effectvely from lne drawngs as from realstc mages [1,3 5]. Furthermore, the 3D percept s usually close to verdcal. We have recently presented a computatonal model that can recover 3D shape of a symmetrc or approxmately symmetrc 3D object from a sngle 2D mage (lne drawng) of ths object. The model does ths by applyng a pror constrants to a 3D nterpretaton of an mage of the object s contours. The a pror constrants ncluded: mrror-symmetry of the 3D shape, planarty of ts contours, maxmum 3D compactness and mnmum surface area of the convex hull of the 3D contours [6 11]. Fgure 1. A 2D closed curve that looks lke a 2D projecton of a 3D mrror-symmetrc curve. See Demo 1 n supplemental materal for an nteractve llustraton of the 3D symmetrc nterpretaton produced from ths 2D curve [2]. A 3D symmetry s a natural pror n recoverng 3D shapes from 2D mages. Most objects n our natural envronment are at least approxmately symmetrc: anmal and human bodes [12], as well as many man-made objects are mrror symmetrc, trees and flowers are rotatonally symmetrc and lmbs and torsos of anmals are characterzed by translatonal symmetry. Clearly, f a vson system of a human or a robot can assume that the object n front of her s symmetrc, the 3D shape recovery becomes much easer. How much easer? Consder mrror-symmetry. Vetter and Poggo [13] showed that a sngle 2D orthographc mage of a mrror-symmetrc 3D shape determnes ths shape wth only one unknown parameter. If a 2D perspectve mage s used, the 3D shape recovery s unque [14 17]. In these models, the contour-confgural organzaton was provded to the model by a human user. By contour-confgural organzaton we mean: fndng contours n the mage, determnng whch contours and features n the mage correspond to symmetrc contours and features n the 3D nterpretaton, whch contours are co-planar and whch contours are on the symmetry plane. Note that what we call contour-confgural organzaton s smlar to the tradtonal phenomenon called
Symmetry 2011, 3 367 perceptual organzaton. The man dfference s that contour-confgural organzaton ncorporates operatons such as establshng symmetry and coplanarty n the 3D nterpretaton. These operatons go beyond the tradtonal perceptual organzaton. We decded to ntroduce ths new concept (followng the suggeston of the Edtor) because we want to emphasze that the processes we nclude are related to the emergence of the percept of the shape of an object, not to arbtrary groupng of features n the mage. Establshng contour-confgural organzaton s natural and easy to a human observer. However, we are stll far from understandng the underlyng mechansms and there s no computer vson algorthm whose performance n establshng contour-confgural organzaton comes even close to that of a human observer. Fgure 2. An asymmetrc par of 2D curves and the 3D symmetrc nterpretaton. (a) An asymmetrc par of 2D curves. These curves can be nterpreted as a 2D perspectve projecton of a symmetrc par of 3D curves whose symmetry plane s slanted at an angle of 40 (see Theorem 1); (b) Three dfferent vews of the 3D symmetrc nterpretaton produced from the par of the 2D curves n (a). The numbers n the bottom are the values of the slant σ s of the symmetry plane of the symmetrc par of the 3D curves. For σ s equal to 40, the mage s dentcal to that n (a). When σ s s 90, ts 2D projecton tself becomes symmetrc. See Demo 2 n supplemental materal for an nteractve llustraton of the 3D symmetrc curves [2]. (a) (b) Front vew Can a symmetry constrant be appled to an mage for whch contour-confgural organzaton has not been establshed? In other words, can 3D symmetry, tself, be used as a tool n establshng contour-confgural organzaton? The answer s, n a general case, negatve. Under qute general assumptons, any par of 2D curves s consstent wth 3D symmetrc nterpretatons. For example, a par of 2D curves n Fgure 2a does not look lke a 2D projecton of a symmetrc par of 3D curves. However, they can be actually nterpreted as a symmetrc par of 3D curves by allowng the degenerate (accdental) vew of the 3D curves (Fgure 2b). Namely, some characterstc features of the 3D symmetrc curves become hdden n the depth drecton (see Dscusson). The reader can see pars of 2D curves (Fgures 1,2,3,5,7,8,12 and 13) and ther 3D symmetrc nterpretatons n our onlne demos [2]. Note that ths paper focuses on the process of constructng or nterpretng, rather than recoverng or re-constructng a 3D shape from a 2D lne drawng. When a 3D shape s beng reconstructed from a lne drawng, t s assumed that ths lne drawng s a 2D projecton produced by some 3D object. For the reconstructon to be accurate, t s necessary to know whether the 2D mage s a result of a perspectve or an orthographc projecton and whether the 3D object was symmetrc, n the
Symmetry 2011, 3 368 frst place. Dependng on the actual projecton type, the reconstructon may or may not be unque. In ths paper, we do not have to know what the actual projecton type s and whether the 3D object was symmetrc. In fact, the 3D object dd not have to exst; the 2D mage could have been drawn by an artst wthout any reference to a 3D object. So, we can assume here the type of projecton and then we can always construct a 3D symmetrc shape. Ths s why we use the word 3D nterpretaton, rather than 3D reconstructon throughout ths paper. As a consequence of constructng, rather than reconstructng, we are not concerned whether the 3D nterpretaton s accurate or not. In partcular, even f the 2D mage was produced by an asymmetrc 3D shape, 3D symmetrc nterpretatons exst. The concept of accuracy s rrelevant here. In Secton 2, t s formally stated and proved that any par of suffcently regular 2D curves can be nterpreted as a symmetrc par of 3D curves. Ths s proved by showng how a symmetrc par of 3D curves s produced from the par of 2D curves. The man dea behnd the proofs s farly smple. Take a par of ponts p and q n the 2D mage. There s always a par of 3D ponts P and Q, whose mages are p and q, such that P and Q are symmetrc wth respect to some plane. The symmetry plane bsects the lne segment P Q and s orthogonal to ths segment. There are nfntely many such solutons. The ndvdual theorems specfy the famly of these solutons for perspectve and orthographc projectons and show that f the mage curves are contnuous, the 3D symmetrc nterpretatons are also contnuous, for both one-to-one and many-to-many pont correspondences. Frst, we provde a proof for a smple case where there s a unque correspondence of pars of symmetrc ponts. We then generalze the theorems to the case of multple correspondences. In Secton 3, we dscuss the role of symmetry n contour-confgural organzaton, as well as the role of other constrants n detectng symmetry from a sngle 2D mage. 2. Theorems and Proofs We begn wth notaton. Consder two curves Φ and Ψ n a 3D space and ther 2D mages φ and ψ. Let Φ and Ψ be symmetrc wth respect to a plane Π s, whose normal s n s (n x, n y, n z ). Let P (x Φ, y Φ, z Φ ), be a pont on Φ and Q (x Ψ, y Ψ, z Ψ ) be ts symmetrc counterpart on Ψ. Symmetry lne segments, whch are lne segments connectng pars of correspondng ponts on Φ and Ψ are parallel to the normal of Π s n the 3D space. Perspectve mages of these lnes ntersect at the vanshng pont v on the mage plane. The 3D orentaton of Π s s specfed by ts slant σ s and tlt τ s. Wthout restrctng generalty, assume τ s = 0. Note that when τ s s not zero, we can always rotate the 3D coordnate system around the z-axs by τ s. Under ths assumpton, the normal to the symmetry plane s n s (n x, 0, n z ). The slant of the symmetry plane s σ s = atan(n x /n z ). In the followng theorems, let z = 0 be the mage plane Π I and the x- and y-axes of the 3D Cartesan coordnate system be the 2D coordnate system on the mage plane. Let the center of projecton F be on the postve sde of the z-axs of the 3D Cartesan coordnate system: z f > 0 where z f s a z-value of F. We assume n ths paper that 2D curves n Π s are fntely long and tame: Defnton 1: A 2D curve s tame when t s connected and composed of a fnte number of C 2 arcs that have followng propertes; each arc s twce contnuously dfferentable and a tangent lne at every non-endpont of the arc does not have any ntersecton wth the arc.
Symmetry 2011, 3 369 Tame curves have fnte number of nflectons and turns. The defnton excludes, for example, pathologcal curves (lke fractal curves), whch have nfntely many nflectons or turns (see [18] for further dscusson). 2.1. A Par of 2D Curves wth Unque Correspondences The Case of a Perspectve Projecton We frst consder the case of a perspectve projecton. The equvalent theorem for an orthographc projecton wll be proved as a specal case of a perspectve projecton. Theorem 1 states that for any par of curves n the 2D mage, there exsts a par of 3D curves that are symmetrc wth respect to a plane. The gst of the proof s as follows. Gven a par of 2D curves, the vanshng pont of a perspectve projecton s computed from the endponts of the curves. The vanshng pont determnes unque pont correspondences between the two curves. It also determnes the symmetry plane unquely for a gven poston of the center of perspectve projecton F. Gven the plane of symmetry, for any par of 2D ponts t s always possble to fnd a par of 3D ponts that are mrror symmetrc wth respect to ths plane. Theorem 1: Let φ and ψ be curves n a 2D mage that are tame. Let the endponts of φ be e φ0 and e φ1, and the endponts of ψ be e ψ0 and e ψ1. Assume that the lnes e φ0 e ψ0 and e φ1 e ψ1 ntersect at a pont v that () s not on φ or ψ and () s not between e φ0 and e ψ0 or between e φ1 and e ψ1. Addtonally, assume that each half lne that emanates from v and ntersects φ has a unque ntersecton wth ψ and vce versa (see Fgure 3). Then, for a gven center of projecton F there exsts a par of contnuous curves Φ and Ψ and a plane Π s n a 3D space such that Φ and Ψ are mrror-symmetrc wth respect to Π s and that φ s a perspectve projecton of Φ and ψ s a perspectve projecton of Ψ. Proof: Fgure 3. F = [0, 0, z f ] s the center of perspectve projecton and Π I (z = 0) s the mage plane. φ and ψ are two gven curves on the mage plane. e φ0 and e φ1 are the endponts of φ, and e ψ0 and e ψ1 are the end ponts of ψ. The lnes e φ0 e ψ0 and e φ1 e ψ1 ntersect at pont v on the x-axs. A lne that s emanatng from v and ntersects wth φ has a unque ntersecton wth ψ and vce versa. y p q v o F z x
Symmetry 2011, 3 370 In order to prove ths theorem, we have to show that for any par of correspondng ponts on φ and ψ, we can fnd ther backprojectons n the 3D space, such that these backprojected ponts are mrror-symmetrc wth respect to the same plane Π s. That s, the lne segment connectng the backprojected ponts s bsected by Π s and parallel to the normal of Π s. It wll be also shown that the backprojected ponts form a par of contnuous curves. Let s set the drecton of x-axs so that the vanshng pont v s on the x-axs, v = [x v, 0, 0]. We express φ and ψ n a polar coordnate system (r, α), where r s the dstance from the vanshng pont v and α s the angle measured relatve to the drecton of the x-axs. Then, the pont p = [x φ, y φ, 0] = [x v + r φ (α )cosα, r φ (α )snα, 0] on φ and the pont q = [x ψ, y ψ, 0] = [x v + r ψ (α )cosα, r ψ (α )snα, 0] on ψ are correspondng. Note that both r φ (α) and r ψ (α) are contnuous functons and they are always postve (r φ (α), r ψ (α) > 0). Let the equaton of the symmetry plane Π s be: t [ a b c][ x y z] = d (1) P and Q, the 3D nverse perspectve projectons of p and q, are symmetrc wth respect to Π s f and only f they satsfy the followng two requrements: the lne segment connectng P and Q s parallel to the normal of Π s and s bsected by Π s. The followng equaton represents the fact that the lne segment connectng P and Q s parallel to the normal of the plane Π s : ( P Q ) [ a b c] = [ 0 0 0] (2) Note that n an nverse perspectve projecton, an mage pont [x, y, 0] projects to a 3D pont [x(z f z)/z f, y(z f z)/z f, z]. Hence, P = [(z f z Φ )(x v + r φ (α )cosα )/z f, (z f z Φ )r φ (α )snα /z f, z Φ ] and Q = [(z f z Ψ )(x v + r ψ (α )cosα )/z f, (z f z Ψ )r ψ (α )snα /z f, z Ψ ]. Then, combnng (1) and (2), we obtan: x z v f d x + z + c From Equaton (3), we obtan the followng three facts. Frst, d/c s an ntersecton of the symmetry plane Π s and the z-axs; t specfes the poston of Π s. Second, the normal to ths plane s [ x v /z f, 0, 1], whch s parallel to a vector [x v, 0, z f ] connectng the center of perspectve projecton wth the vanshng pont. Ths mmedately follows from the fact that v s the vanshng pont correspondng to the lnes connectng the pars of 3D symmetrc ponts, whch are all normal to the symmetry plane. Thrd, a vanshng lne (horzon) h of Π s s parallel to the y-axs on the mage plane Π I. The lne h ntersects x-axs at: = 0 (3) x h z = x 2 f v (4) The next equaton represents the fact that the lne segments connectng pars of 3D symmetrc ponts are bsected by the symmetry plane. Let M be the mdpont between P and Q the mdpont les on the symmetry plane Π s : x x t v t v P + Q 0 1 M = 0 1 z f z f 2 t = d c (5)
Symmetry 2011, 3 371 From Equatons (2) and (5), a perspectve projecton of M to the mage plane Π I can be wrtten as follows: xm m = y m 0 t 2rϕ ( α) rψ ( α) xv + cos ( rϕ ( α) + rψ ( α) ) 2rϕ ( α) rψ ( α ) = snα ( rϕ ( α) + rψ ( α )) 0 t α (6) Equaton (6) shows that m s on a 2D lne segment p q and s determned only by 2D mage features on Π I. It does not depend on the poston of the center of projecton F. Recall that both r φ (α) and r ψ (α) are contnuous functons and they are always postve (r φ (α), r ψ (α) > 0). It follows that the mdponts of the correspondng pars of ponts on φ and ψ form a 2D contnuous curve between φ and ψ on Π I. From Equatons 2 6, we have: z z Φ Ψ = z = z f f + + 2rψ ( α )( z f + d c) xh ( rϕ ( α ) + rψ ( α ))( xm xh ) 2rϕ ( α )( z f + d c) xh ( r ( α ) + r ( α ))( x x ) ϕ It s obvous that (7) and (8) represent contnuous functons unless m s on h: x m = x h. Usng Equatons (6) and (4), we can rewrte x m = x h as follows: r ( α ) ϕ r ( α ) ψ Note that the left-hand sde of Equaton (9) s a cross-rato [x v + r φ (α )cosα, x v + r ψ (α )cosα ; x v, x h ] = [x φ, x ψ ; x v, x h ]. If x m = x h, z Φ and z Ψ dverge to ± and Φ and Ψ are not contnuous. Ths s because a projectng lne emanatng from F and gong through m does not ntersect Π s. As a result, M, whch should be a mdpont between P and Q, cannot be determned. Recall that 2D projectons of mdponts of 3D symmetrc pars of ponts form a 2D contnuous curve on Π I. Hence, ths curve must not have any ntersecton or tangent pont wth h. The whole curve must be ether to the left or rght of h. It follows that the denomnators n (7) and (8) must be always postve or always negatve for gven φ and ψ. If ths crteron s not satsfed, the 3D curves wll not be contnuous. Note that f the poston of the center of projecton F s a free parameter (ths happens when the camera s uncalbrated), t s always possble to set F and thus h so that the crteron for contnuty wll be satsfed because the curve connectng the mdponts does not depend on F. Note that d/c s the only free parameter n Equatons (7) and (8), once the vanshng pont and the center of projecton are fxed. Specfcally, Equatons (7) and (8) show that the left-hand sdes are lnear functons of d/c. Recall that n an nverse perspectve projecton, an mage pont [x, y, 0] projects to a 3D pont [x(z f z)/z f, y(z f z)/z f, z]. Assume that d/c z f otherwse, the 3D nterpretaton wll be degenerate wth all the 3D ponts concdng wth the center of perspectve projecton F (Except for the case when the symmetry plane concdes wth the YOZ plane. Ths can happen when the mage curves are themselves symmetrc). It can be seen that d/c determnes the sze, but not the shape of the ψ ( xv + rψ ( α )cos( α ) xh ) ( x + r ( α )cos( α ) x ) v ϕ m h h = 1 (7) (8) (9)
Symmetry 2011, 3 372 3D curves Φ and Ψ; z f + d/c s a scale factor wth respect to F as a center of scalng. Recall that the denomnators n (7) and (8) must be always postve or always negatve. From Equatons (7) and (8), d/c can be adjusted so that Φ and Ψ are n front of the center of projecton F and the mage plane Π I. The symmetrc par of 3D curves produced from the curves n Fgure 3 usng Equatons (7) and (8) s shown n Fgure 4. Fgure 4. Front and sde vews of a symmetrc par of 3D curves produced from the par of 2D curves n Fgure 3. The dstance z f between the center of projecton and the mage plane, together wth the vanshng pont v determne the slant σ s of the symmetry plane Π s (the slant s 30 n ths case). See Demo 3 n supplemental materal for an nteractve llustraton of the 3D symmetrc curves [2]. Note that a perspectve projecton s used n Demo 3. As a result, the two curves n the sde vew do not project to the same curve on the mage: the farther curve projects to a smaller mage. The sde vew n ths fgure was computed usng an orthographc projecton. As a result, the two symmetrc 3D curves project to the same 2D mage. Front vew Sde vew In ths proof, t was assumed that the poston of the vanshng pont v s known or can be computed from the gven 2D mage. If the poston of the vanshng pont on the mage plane s not known or s uncertan n the 2D mage, the shape of the 3D symmetrc nterpretaton s defned up to two free parameters [19]. These two unknown parameters correspond to the slant and tlt of the symmetry plane Π s. 2.2. A Par of 2D Curves wth Unque Symmetrc Correspondences the Case of an Orthographc Projecton An orthographc projecton s produced from a perspectve projecton by movng the center of perspectve projecton to nfnty. As a result, the vanshng pont correspondng to the symmetry lne segments s also moved to nfnty regardless of the slant of the symmetry plane. Ths mples that the 3D symmetrc nterpretaton s always possble regardless of the poston of the 2D curves on the mage plane. In other words, the crtera for decdng whether the 3D curves are behnd or n front of the camera are rrelevant n the case of an orthographc projecton. We begn wth modfyng Equatons (7) and (8) so that the poston of the vanshng pont s expressed as a functon of the focal length of the camera. It wll then be easy to transform the equatons representng a perspectve projecton to equatons representng an orthographc projecton.
Symmetry 2011, 3 373 Under a perspectve projecton, the projected symmetry lne segments n Π I ntersect at a vanshng pont v. Snce v s an ntersecton of Π I and a lne whch emanates from F and s parallel to n s, the poston of v s [x v, 0, 0] = [ z f tanσ s, 0, 0]. The sne and cosne of the slant σ s of the symmetry plane Π s can be expressed as follows: snσ s = xv L cosσ s = z f L where L s the dstance between the vanshng pont v and the center of projecton F: (10) 2 L = x v + z f 2 (11) Let P = [x Φ, y Φ, z Φ ], be a pont on Φ and Q = [x Ψ, y Ψ, z Ψ ] be ts symmetrc counterpart on Ψ. Recall that the symmetry lne segments are parallel to the normal n s of the symmetry plane Π s and n s = [snσ s, 0, cosσ s ]. Hence, y Φ = y Ψ. Let, p = [x φ, y φ, 0] be a perspectve mage of P and q = [x ψ, y ψ, 0] be a perspectve mage of Q n Π I. A lne segment connectng P and Q s a symmetry lne segment and a lne segment connectng p and q s a projected symmetry lne segment; recall that a projected symmetry lne segment ntersects the x-axs at v. The p and q were represented n a polar coordnate system and wrtten as p = [x φ, y φ, 0] = [x v + r φ cosα, r φ snα, 0] and q = [x ψ, y ψ, 0] = [x v + r ψ cosα, r ψ snα, 0], where r φ and r ψ are the dstances of p and q from v when α = α. Note that the 3D ponts P and Q and ther 2D projectons p and q satsfy Equatons (7) and (8). From Equatons (7), (10) and (11), we obtan z Φ (an analogous formula can be wrtten for z Ψ ): z Φ x = ψ 2d ccosσ s 2xϕ xψ snσ s 1 + + xϕ cos2σ s L L ( xϕ + xψ ) cos2σ s 2xϕ xψ snσ s + sn 2σ s z z L f Recall that an orthographc projecton s produced from a perspectve projecton by movng the center of projecton F to nfnty: z f +. As z f goes to nfnty, L goes to postve nfnty, as well. From Equaton (12), the lmt of z Φ as z f goes to nfnty s: z f lm z Φ x = ψ s f + xϕ cos2σ s d c sn 2σ d c sn 2σ s (12) (13) The lmt of z Ψ s obtaned n an analogous way: lm z z f Ψ x = ϕ + xψ cos2σ s d sn 2σ s c (14) Recall that n an nverse perspectve projecton, an mage pont [x, y,0] projects to a 3D pont [x(z f z)/z f, y(z f z)/z f, z]. As z f goes to nfnty, the lmt of [x(z f z)/z f, y(z f z)/z f, z] s [x, y, z], whch s an nverse orthographc projecton of [x, y, 0]. Note that x v goes to negatve nfnty as z f goes to nfnty; t follows that all α become zero. Ths means that the projected symmetry lne segments become parallel to one another and to the x-axs, and the vanshng pont v goes to nfnty. Hence, the slant σ s of the symmetry plane cannot be computed from the 2D mage; nstead, σ s becomes a free parameter n the 3D nterpretaton under an orthographc projecton. It follows that there are nfntely many 3D symmetrc curves that are consstent wth a par of 2D curves φ and ψ. In other words, the 3D curves form a one-parameter
Symmetry 2011, 3 374 famly characterzed by σ s ; σ s changes the aspect rato and the orentaton of the 3D shapes of the curves Φ and Ψ [10]. Note that f sn2σ s = 0 (σ s s 0 or 90 ), z Φ and z Ψ dverge to ±. Hence, σ s should not be 0 or 90. These two cases correspond to degenerate vews of Φ and Ψ. When σ s s 0, the symmetry plane s parallel to the mage plane; φ and ψ wll then concde wth each other n the 2D mage. In such a case, the 3D recovery of a par of symmetrc 3D curves becomes trval: one produces any Φ from ϕ and then Ψ s obtaned as a mrror reflecton of Φ. When σ s s 90, the symmetry plane s perpendcular to the mage plane. In such a case, φ and ψ, themselves, must be mrror symmetrc n the 2D mage n order for the 3D symmetrc nterpretaton to exst. But then, the 2D curves themselves represent one possble 3D symmetrc nterpretaton. The rato d/c s another free parameter, but t only changes the poston of Φ and Ψ along the z-axs and does not change ther 3D shapes or orentatons. From these results, Theorem 1 for a perspectve projecton generalzes to Theorem 2 for an orthographc projecton. Theorem 2: Let φ and ψ be curves that are tame n a sngle 2D mage. Let the endponts of φ be e φ0 and e φ1, and the endponts of ψ be e ψ0 and e ψ1. Assume that φ and ψ have the followng propertes: () e φ0 e ψ0 e φ1 e ψ1 and () a lne that s parallel to e φ0 e ψ0 and ntersects wth φ has a unque ntersecton wth ψ and vce versa (see Fgure 5). Then, there exst nfntely many pars of contnuous curves Φ and Ψ and a plane Π s n a 3D space, such that Φ and Ψ are mrror-symmetrc wth respect to Π s and that φ s an orthographc projecton of Φ and ψ s an orthographc projecton of Ψ. Proof: Fgure 5. φ and ψ are two 2D curves. e φ0 and e φ1 are the endponts of φ, and e ψ0 and e ψ1 are the end ponts of ψ. The lnes e φ0 e ψ0 and e φ1 e ψ1 are parallel to the x-axs and do not have any ntersecton wth φ and ψ. A lne that s parallel to e φ0 e ψ0 and ntersects wth φ has a unque ntersecton wth ψ and vce versa. y p q x
Symmetry 2011, 3 375 Let the orentatons of the lne segments e φ0 e ψ0 and e φ1 e ψ1 be horzontal. Ths does not restrct the generalty: f these lne segments are not horzontal, we rotate the mage so that they become horzontal. For any pont p = [x φ, y φ ] on φ, we fnd ts counterpart on ψ as q = [x ψ, y ψ ]. Note that q s found as an ntersecton of ψ and a horzontal lne y = y φ. Hence, y φ = y ψ. We assume that ths ntersecton s unque. Then, both φ and ψ can be represented as functons of y: x φ = x φ (y φ ) and x ψ = x ψ (y φ ). From Equatons (13) and (14), the 3D symmetrc curves Φ and Ψ are produced by computng the postons of all ponts P and Q as follows: xψ ( yϕ ) + xϕ ( yϕ )cos2σ s P = xϕ ( yϕ ) yϕ d c sn 2σ s xϕ ( yϕ ) + xψ ( yϕ )cos2σ s Q = xψ ( yϕ ) yϕ d c sn 2σ s where σ s s a slant of the symmetry plane. The tlt τ s of the symmetry plane s zero. Equatons (15) and (16) allow one to compute a par of 3D symmetrc curves Φ and Ψ from a par of 2D curves φ and ψ under an orthographc projecton. It s obvous from these equatons that Φ and Ψ are contnuous when φ and ψ are contnuous. Recall that slant σ s s a free parameter; t can be arbtrary, except for sn2σ s = 0. So, the 3D symmetrc curves form a one-parameter famly characterzed by σ s. Equatons (15 16) show that a relaton between the par of 2D curves φ and ψ and the par of 3D curves Φ and Ψ becomes computatonally smple when σ s s 45 (or 45 ) and d/c s 0; the absolute value of the z-coordnate of Q s equal to the x-coordnate of P and the absolute value of the z-coordnate of P s equal to the x- coordnate of Q. The symmetrc par of 3D curves produced usng Equatons (15) and (16) and consstent wth φ and ψ n Fgure 5 s shown n Fgure 6. If the drecton of the lnes connectng the correspondng ponts on φ and ψ s not known or s uncertan, the famly of 3D nterpretatons s characterzed by two parameters: slant and tlt of the symmetry plane. Fgure 6. Vews of a symmetrc par of 3D curves produced from the par of 2D curves n Fgure 5. The slant σ s of the symmetry plane Π s was set to 45. See Demo 4 n supplemental materal for an nteractve llustraton of the 3D symmetrc curves [2]. (15) (16) Front vew Sde vew
Symmetry 2011, 3 376 2.3. A Par of 2D Curves wth Multple Symmetrc Correspondences In the two theorems above, t was assumed that correspondences between ponts of φ and ψ are unque. We generalze these theorems to the case of non-unque correspondences. A pont on φ can have multple correspondng ponts on ψ (and vce versa). In such a case, the 3D nterpretaton of φ wll have segments whose 2D projectons perfectly overlap one another n the 2D mage (Fgure 7). In other words, the 3D symmetry wll be hdden n the depth drecton, and thus, the 3D vew wll be degenerate (see Dscusson). Frst, we consder the case of an orthographc projecton. Ths case wll then be generalzed to a perspectve projecton. Fgure 7. An asymmetrc par of 2D curves wth multple symmetrc correspondences and ts 3D symmetrc nterpretaton. (a) An asymmetrc par of 2D curves. Some ponts of the rght curve correspond to three ponts of the left curve. These curves can be stll nterpreted as a 2D orthographc projecton of a symmetrc par of 3D curves. The slant σ s of the symmetry plane was set to 35 ; (b) Three dfferent vews of the 3D symmetrc nterpretaton produced from the par of the 2D curves n (a). The numbers n the bottom are the values of the slant σ s of the symmetry plane of the symmetrc par of the 3D curves. For σ s equal to 35, the mage s dentcal to that n (a). When σ s s 90, ts 2D projecton tself becomes symmetrc. See Demo 5 n supplemental materal for an nteractve llustraton of the 3D symmetrc curves [2]. (a) (b) Front vew Theorem 3: Let φ and ψ be curves that are tame n a 2D mage. Let the endponts of φ be e φ0 and e φ1, the endponts of ψ be e ψ0 and e ψ1. Let a lne connectng e φ0 and e ψ0 be l 0 and that connectng e φ1 and e ψ1 be l 1. Assume that φ and ψ have the followng propertes: () l 0 l 1, () l 0 and l 1 do not have any ntersecton wth φ and ψ and () a lne that s parallel to l 0 and ntersects φ has one or a fnte number of ntersectons wth ψ and vce versa (see Fgure 8). Then, there exst nfntely many pars of contnuous curves Φ and Ψ and a plane Π s n a 3D space, such that Φ and Ψ are mrror-symmetrc wth respect to Π s and that φ s an orthographc projecton of Φ and ψ s an orthographc projecton of Ψ.
Symmetry 2011, 3 377 Proof: Fgure 8. φ and ψ are 2D curves. e φ0 and e φ1 are the endponts of φ, and e ψ0 and e ψ1 are the end ponts of ψ. l 0 s a lne connectng e φ0 and e ψ0, and l 1 s a lne connectng e φ1 and e ψ1. l 0 and l 1 are parallel to the x-axs and do not have any ntersecton wth φ and ψ. A lne that s parallel to l 0 and ntersects wth φ has one or more ntersectons wth ψ and vce versa. p 5 p 15 q 13 p 4 p 6 p 8 p 14 q 8 q 12 p 3 p 7 p 9 p 13 q 3 q 7 q 9 q 11 p 2 p 10 p 12 q 2 q 4 q 6 q 10 p 1 p 11 q 1 q 5 In order to prove ths theorem, we frst dvde a par of φ and ψ nto multple pars of fragments, such that each par satsfes the assumptons of Theorem 2. Then, we fnd ther backprojectons that are mrror-symmetrc pars of contnuous curves n the 3D space. Next, we wll show that these multple pars of 3D curves can share a common symmetry plane Π s and produce a symmetrc par of contnuous curves. As before we assume that the orentatons of the lne segments e φ0 e ψ0 and e φ1 e ψ1 are horzontal. In ths case, e φ0, e φ1, e ψ0 and e ψ1 are ether global maxma or mnma of φ and ψ along a vertcal axs on the 2D mage. Consder horzontal lnes that are tangent to the curves at ther local extrema (Fgure 8). Intersectons and tangent ponts of these horzontal lnes wth φ and ψ are labeled by numbers sequentally along each curve; p 1,, p m on φ and q 1,, q n on ψ (n Fgure 8, n = 15, and m = 13). Both φ and ψ are dvded nto segments p 1 p 2, p 2 p 3 etc. Let the endponts e φ0, e φ1, e ψ0 and e ψ1 on φ and ψ be p 0, p m+1, q 0 and q n+1, respectvely. Let c φ be a segment of ϕ connectng p and ts successor p +1. Then, c φ s a curve that s monotonc and contnuous between p and p +1 and these two ponts are the endponts of c φ. Smlarly, let c ψj be a segment of ψ connectng q j and ts successor q j+1. A segment c φ of φ has, at least, one correspondng segment of ψ; the endponts of these segments of ϕ and ψ form parallel lne segments. From Theorem 2, a par of c φ and each of the correspondng segments of ψ are consstent wth an nfnte number of 3D symmetrc nterpretatons under an orthographc projecton; the one-parameter famly of symmetrc pars of 3D curves s characterzed by the slant of a symmetry plane. The tlt of the symmetry plane s zero and ts depth along the z-axs s arbtrary. It follows that, among all correspondng pars of 2D segments of these curves, ther possble 3D symmetrc nterpretatons can share a common symmetry plane Π s wth some slant σ s and depth. Hence, φ and ψ
Symmetry 2011, 3 378 are consstent wth a one parameter famly of symmetrc pars of the 3D fragmented curves that are backprojectons of the 2D segments of the 2D curves. In order to prove Theorem 3, we show that, for each member of the famly, the symmetrc pars of the 3D fragmented curves produce a symmetrc par of 3D contnuous curves whose endponts are backprojectons of the endponts of φ and ψ. An orthographc projecton of such a symmetrc par of the 3D contnuous curves wll be φ and ψ. A table representng pars of the segments of φ and ψ and ther endponts s shown n Fgure 9. The rows of the table represent the ponts p 0,, p m+1 on φ. The columns represent the ponts q 0,, q n+1 on ψ. A crcle at (p, q j ) represents a correspondng par of ponts p and q j n Fgure 8. These crcles are nodes n a graph representng the possble correspondences among pars of ponts n Fgure 8. Let an edge n ths graph connectng (p, q j ) and (p k, q l ) be labeled (p, q j )-(p k, q l ). Note that the edge (p, q j )-(p k, q l ) s the same as (p k, q l )-(p, q j ). The edge (p k, q l )-(p, q j ) represents a par of segments of ϕ and ψ, such that c φ mn(,k) connects p and p k and c ψ mn(j,l) connects q j and q l. Recall that the endponts of each segment on ϕ and ψ are two neghborng ponts of φ or ψ; I k = 1 and j l = 1. Hence, an edge n the graph shown n Fgure 9 can only connect two nodes that are dagonally next to each other n the table and a node can be connected to, at most, four nodes by four edges. A par of nodes can be connected to each other only by a sngle edge. The end nodes of (p, q j )-(p k, q l ) are ether the maxma or mnma of these segments and ther endponts form horzontal lne segments. Hence, from the Theorem 2, a par of segments of ϕ and ψ represented by each edge n the graph s consstent wth a one-parameter famly of pars of 3D curves that s symmetrc wth respect to the common symmetry plane Π s. Fgure 9. Ths table represents pars of segments and pars of ponts on of ϕ and ψ. The rows represent the ponts p 0,, p m+1 on φ. The columns represent the ponts q 0,, q n+1 on ψ. A crcle at (p, q j ), whch s a node n the graph, represents a correspondng par of ponts p and q j ; p and q j form a horzontal lne segment n Fgure 8. The top left and bottom rght nodes represent pars of endponts of φ and ψ. An edge (p, q j )-(p k, q l ) (or (p k, q l )-(p, q j )) connectng (p, q j ) and (p k, q l ) represents a par of segments of ϕ and ψ. p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 10 p 11 p 12 p 13 p 14 p 15 p 16 q 0 q 1 q 2 q 3 q 4 q 5 q 6 q 7 q 8 q 9 q 10 q 11 q 12 q 13 q 14
Symmetry 2011, 3 379 Consder two edges (p, q j )-(p k, q l ) and (p, q j )-(p g, q h ) connected to a common node (p, q j ). These edges represent two pars of segments of the 2D curves; a par c φmn(,k) and c ψmn(j,l) and a par c φmn(,g) and c ψmn(j,h). The two segments c φmn(,k) and c φmn(,g) are connected to each other at ther common endpont p (see Fgure 8). In the same way, c ψmn(j,l) and c ψmn(j,h) are connected to each other at ther common endpont q j. Hence, these two pars of 2D curves can be regarded as a par of 2D curves whose endponts are represented by (p k, q l ) and (p g, q h ) that are the end nodes of the path formed by the edges. Note that each (p, q j )-(p k, q l ) and (p, q j )-(p g, q h ) s consstent wth nfntely many pars of 3D contnuous curves. Assume that they are symmetrc wth respect to the common symmetry plane Π s whose slant and depth are gven. Then, two symmetrc pars of 3D contnuous curves are unquely determned; these 3D curves are backprojectons of c φmn(,k), c φmn(,g), c ψmn(j,l) and c ψmn(j,h), respectvely. The 3D curves that are backprojectons of c φmn(,k) and c φmn(,g) are connected to each other at ther common endpont that s a backprojecton of p ; these two 3D curves can be regarded as a sngle 3D curve. The same way, the 3D curves that are backprojectons of c ψmn(j,l) and c ψmn(j,h) can be regarded as a sngle 3D curve. It follows that the two symmetrc pars of 3D curves produced from c φmn(,k), c φmn(,g), c ψmn(j,l) and c ψmn(j,h) can be regarded as a sngle symmetrc par of 3D contnuous curves. Ther endponts are backprojectons of the 2D ponts represented by the end nodes (p k, q l ) and (p g, q h ) of the contnuous path. Ths can be generalzed to all segments of ϕ and ψ as follows. A contnuous path of the edges n the table n Fgure 9 represents a par of 2D contnuous curves; the 2D curves are composed of the 2D segments of φ and ψ and ther endponts are represented by the end nodes of the path. The 2D curves are consstent wth an nfnte number of symmetrc pars of 3D contnuous curves. The endponts of the 3D curves are backprojectons of the ponts represented by the end nodes of the path. So, f there s a contnuous path connectng (p 0, q 0 ) and (p m+1, q n+1 ) n the graph n Fgure 9, then ths path represents a par of 2D contnuous curves φ and ψ and ths par of the 2D curves s consstent wth a one-parameter famly of symmetrc pars of 3D contnuous curves. The exstence of a contnuous path of edges connectng (p 0, q 0 ) and (p m+1, q n+1 ) n the graph wll now be proved by usng concepts from a graph theory. A graph s called connected f there s a contnuous path of edges between every par of nodes n the graph. If (p 0, q 0 ) and (p m+1, q n+1 ) belong to a connected graph, there s a path connectng (p 0, q 0 ) and (p m+1, q n+1 ). A connected graph has even number of nodes of odd degree, where degree of a node refers to number of edges connected to the node [20]. If (p 0, q 0 ) and (p m+1, q n+1 ) are the only nodes of odd degree n the graph, they must belong to the same connected graph and there must be a contnuous path connectng (p 0, q 0 ) and (p m+1, q n+1 ). Next, we provde a classfcaton of possble nodes n the graph and show that there are only four types: of degree zero, one, two or four. Ths wll conclude the proof. Consder p 1,, p m on φ and q 1,, q n on ψ. If a node (p, q j ) exsts n the table, p and q j form a horzontal lne segment n Fgure 8. Note that (p, q j ) can be connected to, at most, four nodes n the graph that are dagonally next to (p, q j ): (p 1, q j 1 ), (p 1, q j+1 ), (p +1, q j 1 ) and (p +1, q j+1 ). Hence, the maxmum degree of each node n the table s four. These four neghborng nodes represent possble pars of neghborng ponts of p along φ and q j along ψ. Consder a par of the neghborng ponts p +1 and q j + 1. The node (p +1, q j+1 ) exsts f and only f p +1 and q j+1 are a correspondng par; they form a horzontal lne segment n Fgure 8. Note that p + 1 and p are connected by c φ and q j+1 and q j are connected by c ψj. If both (p +1, q j+1 ) and (p, q j ) exst n the graph, they are connected by
Symmetry 2011, 3 380 (p +1, q j+1 )-(p, q j ) representng a par of segments c φ and c ψj. Therefore, the number of edges n the graph connected to (p, q j ) can be computed by verfyng the exstence of the four neghborng nodes. In order to compute the number of edges connected to each node, ponts and correspondng pars of ponts represented by the nodes n the graph are classfed. Consder ponts p 1,, p m and q 1,, q n. Frst, these ponts can be classfed nto three types: local maxma, local mnma and ponts at whch the curve s monotonc (Fgure 10). If a pont s a local maxmum, ts neghborng ponts are lower than the local maxmum. If a pont s a local mnmum, ts neghborng ponts are hgher than the local mnmum. If a pont s a monotonc pont, one of ts neghborng ponts s hgher and the other s lower. Next, based on the classfcaton of the ponts, the correspondng pars of the ponts can be classfed nto three types (Fgure 11). Type (): f two monotonc ponts are correspondng, a node representng ths par of ponts s connected to two nodes by two edges. Hence, the degree of ths node s two; Type (): f a monotonc pont and a local maxmum/mnmum are correspondng, the degree of a node representng ths par s also two; Type (): f two local maxma/mnma are correspondng, the degree of a node representng ths par s four; Type (v): f a local maxmum and a local mnmum are correspondng, the degree of a node representng ths par s zero. From these facts, the degree of any node whch does not represent the endponts of the two curves s always even (0, 2 or 4). Next, consder pars of endponts of φ and ψ: (p 0, q 0 ) and (p m + 1, q n+1 ). Recall that p 0 = e φ0, p m+1 = e φ1, q 0 = e ψ0 and q n+1 = e ψ1 ; so, the lne segments p 0 q 0 and p m+1 q n+1 are horzontal n Fgure 8. Hence, these pars of the endponts are correspondng pars and the nodes representng these pars exst n the graph. Note that these endponts are global maxma and mnma of φ and ψ; the global maxma form a correspondng par and the global mnma form a correspondng par. Recall that f two local maxma or two local mnma form a correspondng par, there are four correspondng pars of ther neghborng ponts. However, each endpont has only one neghborng pont. Hence, there s one correspondng par of the neghborng ponts for each par of the endponts. Therefore, each (p 0, q 0 ) and (p m+1, q n+1 ) s connected to a sngle node and ther degrees are one, whch s an odd number. Fgure 10. Three types of ponts (sold crcles) on a 2D curve and ther neghborng ponts (open crcles). (a) Local maxmum (b) Local mnmum (c) Monotonc pont
Symmetry 2011, 3 381 Fgure 11. Four types of pars of ponts (sold crcles) and ther neghborng ponts (open crcles). See text for more nformaton. Type () or Type () or Type () Type (v) Note that the case where φ or ψ has a local extremum at whch a horzontal lne connectng e φ0 and e ψ0 or e φ1 and e ψ1 s tangent to the curve of the extremum, s analogous to Type () n Fgure 11. The extremum of the curve corresponds to an endpont of the other curve that s on the horzontal tangent lne. Unlke Type (), the endpont has only one neghborng pont that forms a correspondng par wth the neghborng ponts of the local extremum. Hence, the degree of a node representng the par of the local extremum and the endpont s also two, whch s an even number. From these facts, there are two nodes (p 0, q 0 ) and (p m+1, q n+1 ) whose degrees are odd (1) and the degrees of all other nodes are even (0, 2 or 4). Hence, (p 0, q 0 ) and (p m+1, q n+1 ) must belong to the same connected graph and these two nodes are connected by a contnuous path of edges. Ths contnuous path n the graph represents the correspondences between 3D contnuous curves of a symmetrc par whose orthographc projectons are the 2D curves φ and ψ, respectvely. Once the correspondences are formed, the par of the 3D curves can be produced usng Equatons (15) and (16). Note that the slant of the symmetry plane Π s s a free parameter of the 3D symmetrc nterpretaton of φ and ψ under an orthographc projecton. A symmetrc par of 3D curves produced from the par of 2D curves n Fgure 8 s shown n Demo 6 n supplemental materal [2]. Ths proof of Theorem 3 for an orthographc projecton can be easly generalzed to the case of a perspectve projecton: Theorem 4: Let φ and ψ be curves that are tame n a sngle 2D mage. Let the endponts of φ be e φ0 and e φ1, and the endponts of ψ be e ψ0 and e ψ1. Let a lne connectng e φ0 and e ψ0 be l 0 and that connectng e φ1 and e ψ1 be l 1. Assume that φ and ψ have the followng propertes: () l 0 and l 1 ntersect at a pont v that s not on φ or ψ; () l 0 and l 1 do not have any ntersecton wth φ and ψ; () v s not
Symmetry 2011, 3 382 between e φ0 and e ψ0 or between e φ1 and e ψ1 ; and (v) a half lne that emanates from v and ntersects wth φ has one or a fnte number of ntersectons wth ψ and vce versa. Then, there exsts a par of contnuous curves Φ and Ψ and a plane Π s n a 3D space for a gven center of projecton F, such that Φ and Ψ are mrror-symmetrc wth respect to Π s and that φ s a perspectve projecton of Φ and ψ s a perspectve projecton of Ψ. Proof: In the proof of Theorem 3 for the case of an orthographc projecton, the 2D curves φ and ψ were dvded by lnes whch were parallel to e φ0 e ψ0 and were tangent to ether of the 2D curves. In the case of a perspectve projecton, φ and ψ are dvded by lnes whch emanate from the vanshng pont v and are tangent to ether of the 2D curves. The rest of ths proof s dentcal to that of Theorem 3. The only dfference s that n the case of a perspectve projecton the 3D nterpretaton s unque the slant of the symmetry plane s not a free parameter. In the four theorems above t was assumed that an endpont of one curve corresponds to an endpont of the other curve. Next, we generalze Theorems 3 and 4 to the case where an endpont of a curve may or may not correspond to an endpont of the other curve. Ths can happen, for example, n the presence of occluson. We begn wth the case of an orthographc projecton. Theorem 5: Let φ and ψ be curves that are tame n a 2D mage. Assume that there exst two lnes l 0 and l 1 whch satsfy the followng propertes: () l 0 and/or l 1 s ether tangent to both φ and ψ or passes through ther endponts, () l 0 l 1, () l 0 and l 1 do not have any ntersecton wth φ and ψ and (v) a lne that s parallel to l 0 and ntersects φ has one or a fnte number of ntersectons wth ψ and vce versa (see Fgure 12). Then, there exst nfntely many pars of contnuous curves Φ and Ψ and a plane Π s n a 3D space, such that Φ and Ψ are mrror-symmetrc wth respect to Π s and that φ s an orthographc projecton of Φ and ψ s an orthographc projecton of Ψ. In order to prove ths theorem, we frst extend φ and ψ and obtan a par of 2D curves φ and ψ that perfectly overlap φ and ψ n the 2D mage and satsfy the assumptons of Theorem 3. Then, we fnd the backprojectons of φ and ψ n the 3D space, such that these backprojected curves are mrror-symmetrc wth respect to a plane Π s. Ther orthographc projectons n the 2D mage concde wth φ and ψ, as well as wth φ and ψ. Let the orentatons of l 0 and l 1 be horzontal. In ths case, tangent ponts of φ and ψ to l 0 or l 1 are ether global maxma or mnma of φ and ψ along a vertcal axs on the 2D mage. Assume that the endponts of these curves are not global extrema the case when they are global extrema has been consdered n the prevous theorems. Let the tangent ponts of φ to l 0 and l 1 be t φ0 and t φ1. Let an endpont of φ, whch s closer (as measured by the arc length along the curve) to t φ0 be e φ0, and that, whch s closer to t φ1 be e φ1. The same way, let the tangent ponts of ψ to l 0 or l 1 be t ψ0 and t ψ1, and the endponts ψ be e ψ0 and e ψ1. We extend the 2D curves φ and ψ by addng arcs that start from e φ0, e φ1, e ψ0 and e ψ1, whch are endponts of φ and ψ. The extenson startng from e φ0 s dentcal to the segment of φ between e φ0 and t φ0. Smlarly, the extenson startng from e φ1 s dentcal to the segment of φ between e φ1 and t φ1. Let ths new curve be φ. Let the endponts of φ be e φ0 and e φ1, so that the postons of e φ0 and e φ1 are respectvely the same as those of t φ0 and t φ1 n the 2D mage. The same way, let the extended curve produced from ψ be ψ and the endponts of ψ be e ψ0 and e ψ1. The postons of e ψ0 and e ψ1 are
Symmetry 2011, 3 383 respectvely the same as those of t ψ0 and t ψ1 n the 2D mage. The curves φ and ψ perfectly overlap φ and ψ n the 2D mage (Fgure 12). Therefore, the 3D nterpretatons of φ and ψ are also consstent wth the 3D nterpretatons of φ and ψ. It s easy to see that φ and ψ satsfy the assumptons of Theorem 3. Therefore, t follows from the proof of Theorem 3 that there exsts a one-parameter famly of symmetrc pars of contnuous 3D curves Φ and Ψ, such that φ s an orthographc projecton of Φ and ψ s an orthographc projecton of Ψ. Ths, n turn, mples that φ and ψ are also orthographc projecton of Φ and Ψ. A symmetrc par of 3D curves produced from the par of 2D curves n Fgure 12 s shown n Demo 7 n supplemental materal [2] (It looks lke each of the 3D curves n Demo 6 has four endponts, rather than two, as would be expected from Theorem 3. It also looks lke each of the 3D curves has two bfurcatons. All of the four ponts that look lke endponts are actually 180 turns of the 3D curve. The real endponts of the 3D curve are at the bfurcatons). Proof: Fgure 12. φ and ψ (red sold curves) are 2D curves. e φ0 and e φ1 are the endponts of φ, and e ψ0 and e ψ1 are the end ponts of ψ. l 0 s tangent to both φ and ψ at t φ0 and t ψ0, and l 1 s tangent to both φ and ψ at t φ1 and t ψ1. l 0 and l 1 are parallel to the x-axs and do not have any ntersecton wth φ and ψ. The two curves, φ and ψ are extended by addng arcs (blue dotted curves) that start from e φ0, e φ1, e ψ0 and e ψ1. The addtonal arcs are dentcal to the segments of φ and ψ (note that the blue dotted curves are perfectly overlappng the red sold curves n the mage). They end at e φ0, e φ1, e ψ0 and e ψ1 whose postons are respectvely the same as those of t φ0, t φ1, t ψ0 and t ψ1. See Demo 7 n supplemental materal for an nteractve llustraton of the 3D symmetrc curves produced from φ and ψ [2]. l 1 l 0
Symmetry 2011, 3 384 Ths proof of Theorem 5 for an orthographc projecton can be easly generalzed to the case of a perspectve projecton, the same way as the proof of Theorem 3 was generalzed to the case of perspectve projecton: Theorem 6: Let φ and ψ be curves that are tame n a 2D mage. Assume that there exst two lnes l 0 and l 1 whch satsfy the followng propertes: () l 0 and/or l 1 s ether tangent to both φ and ψ or passes through ther endponts; () l 0 and l 1 ntersect at a pont v that s not on φ or ψ; () l 0 and l 1 do not have any other ntersectons wth φ and ψ; (v) v s not between e φ0 and e ψ0 or between e φ1 and e ψ1, and (v) a half lne that emanates from v and ntersects wth φ has one or a fnte number of ntersectons wth ψ and vce versa. Then, there exsts a par of contnuous curves Φ and Ψ and a plane Π s n a 3D space for a gven center of projecton F, such that Φ and Ψ are mrror-symmetrc wth respect to Π s and that φ s a perspectve projecton of Φ and ψ s a perspectve projecton of Ψ. 3. Dscusson In ths paper, we showed that any par of 2D curves that are suffcently regular s consstent wth a 3D symmetrc nterpretaton under qute general assumptons. We derved the equatons that allow one to compute the 3D curves for the case of perspectve and orthographc projectons. Although the man part of the proofs s farly straghtforward, the result s surprsng and has mportant mplcatons for theores of shape percepton. Consder the examples n Fgures 1, 2, 3, 5, 7, 8 and 13. Although these pars of 2D curves do not look lke symmetrc pars of 3D curves, they do have 3D symmetrc nterpretatons. What s the nature of the process that can produce 3D symmetry from an arbtrary 2D mage? The key element of ths process seems to be related to the concept of degenerate vews. Essentally, the 3D vewng drecton for whch the 3D symmetrc curves project to the gven 2D asymmetrc curves s degenerate and the 3D symmetry becomes hdden n the 2D mage. There are two cases representng a degenerate vew. The frst, more obvous case, happens when multple segments of a 3D curve project to the same segment of a 2D curve. Ths can be seen n the 3D symmetrc nterpretatons of Fgures 7, 8 and 13 (see Demos 5 7 [2]). The second, more subtle case, corresponds to the stuaton where a local curvature of a 3D curve dsappears n the 2D projecton. The smplest example s when a planar curve n the 3D space projects to a straght lne segment n the 2D mage [21]. However, f a curve n the 3D space s not planar, then ts 2D projecton s never a straght lne segment, even for degenerate vews. Recall that curvature of a 3D curve (also called frst curvature) represents the change of the tangent to the curve wthn the plane of the crcle of curvature, whle torson (also called second curvature) represents the change of the tangent to the curve away from the plane of the crcle of curvature [22]. If the normal of the plane of the crcle of curvature s perpendcular to the lne of sght, the change of the tangent wthn ths plane (.e., curvature of the 3D curve) dsappears n the 2D projecton, but the departure from ths plane (.e., torson of the 3D curve) does not. In a sense, for such vews, the local curvature of a 2D curve s a projecton of the local torson of a 3D curve. Fgures 2, 3 and 5 llustrate the second case of degenerate vews (see Demos 2 4 [2]). Apparently, the vsual system rejects 3D symmetrc nterpretatons f they mply degenerate vews. Ths makes sense because degenerate vews are unlkely.